Theorem dilsetN 40616

Description: The set of dilations for a fiducial atom 𝐷. (Contributed by NM, 4-Feb-2012.) (New usage is discouraged.)

Hypotheses

Ref	Expression
dilset.a	⊢ 𝐴 = (Atoms‘𝐾)
dilset.s	⊢ 𝑆 = (PSubSp‘𝐾)
dilset.w	⊢ 𝑊 = (WAtoms‘𝐾)
dilset.m	⊢ 𝑀 = (PAut‘𝐾)
dilset.l	⊢ 𝐿 = (Dil‘𝐾)

Assertion

Ref	Expression
dilsetN	⊢ ((𝐾 ∈ 𝐵 ∧ 𝐷 ∈ 𝐴) → (𝐿‘𝐷) = {𝑓 ∈ 𝑀 ∣ ∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝐷) → (𝑓‘𝑥) = 𝑥)})

Distinct variable groups:   𝑥,𝑓,𝐾   𝑓,𝑀   𝑥,𝑆   𝐷,𝑓,𝑥

Allowed substitution hints:   𝐴(𝑥,𝑓)   𝐵(𝑥,𝑓)   𝑆(𝑓)   𝐿(𝑥,𝑓)   𝑀(𝑥)   𝑊(𝑥,𝑓)

Proof of Theorem dilsetN

Dummy variable 𝑑 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dilset.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
2		dilset.s	. . . 4 ⊢ 𝑆 = (PSubSp‘𝐾)
3		dilset.w	. . . 4 ⊢ 𝑊 = (WAtoms‘𝐾)
4		dilset.m	. . . 4 ⊢ 𝑀 = (PAut‘𝐾)
5		dilset.l	. . . 4 ⊢ 𝐿 = (Dil‘𝐾)
6	1, 2, 3, 4, 5	dilfsetN 40615	. . 3 ⊢ (𝐾 ∈ 𝐵 → 𝐿 = (𝑑 ∈ 𝐴 ↦ {𝑓 ∈ 𝑀 ∣ ∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝑑) → (𝑓‘𝑥) = 𝑥)}))
7	6	fveq1d 6837	. 2 ⊢ (𝐾 ∈ 𝐵 → (𝐿‘𝐷) = ((𝑑 ∈ 𝐴 ↦ {𝑓 ∈ 𝑀 ∣ ∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝑑) → (𝑓‘𝑥) = 𝑥)})‘𝐷))
8		fveq2 6835	. . . . . . 7 ⊢ (𝑑 = 𝐷 → (𝑊‘𝑑) = (𝑊‘𝐷))
9	8	sseq2d 3955	. . . . . 6 ⊢ (𝑑 = 𝐷 → (𝑥 ⊆ (𝑊‘𝑑) ↔ 𝑥 ⊆ (𝑊‘𝐷)))
10	9	imbi1d 341	. . . . 5 ⊢ (𝑑 = 𝐷 → ((𝑥 ⊆ (𝑊‘𝑑) → (𝑓‘𝑥) = 𝑥) ↔ (𝑥 ⊆ (𝑊‘𝐷) → (𝑓‘𝑥) = 𝑥)))
11	10	ralbidv 3161	. . . 4 ⊢ (𝑑 = 𝐷 → (∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝑑) → (𝑓‘𝑥) = 𝑥) ↔ ∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝐷) → (𝑓‘𝑥) = 𝑥)))
12	11	rabbidv 3397	. . 3 ⊢ (𝑑 = 𝐷 → {𝑓 ∈ 𝑀 ∣ ∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝑑) → (𝑓‘𝑥) = 𝑥)} = {𝑓 ∈ 𝑀 ∣ ∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝐷) → (𝑓‘𝑥) = 𝑥)})
13		eqid 2737	. . 3 ⊢ (𝑑 ∈ 𝐴 ↦ {𝑓 ∈ 𝑀 ∣ ∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝑑) → (𝑓‘𝑥) = 𝑥)}) = (𝑑 ∈ 𝐴 ↦ {𝑓 ∈ 𝑀 ∣ ∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝑑) → (𝑓‘𝑥) = 𝑥)})
14	4	fvexi 6849	. . . 4 ⊢ 𝑀 ∈ V
15	14	rabex 5277	. . 3 ⊢ {𝑓 ∈ 𝑀 ∣ ∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝐷) → (𝑓‘𝑥) = 𝑥)} ∈ V
16	12, 13, 15	fvmpt 6942	. 2 ⊢ (𝐷 ∈ 𝐴 → ((𝑑 ∈ 𝐴 ↦ {𝑓 ∈ 𝑀 ∣ ∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝑑) → (𝑓‘𝑥) = 𝑥)})‘𝐷) = {𝑓 ∈ 𝑀 ∣ ∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝐷) → (𝑓‘𝑥) = 𝑥)})
17	7, 16	sylan9eq 2792	1 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝐷 ∈ 𝐴) → (𝐿‘𝐷) = {𝑓 ∈ 𝑀 ∣ ∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝐷) → (𝑓‘𝑥) = 𝑥)})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  {crab 3390   ⊆ wss 3890   ↦ cmpt 5167  ‘cfv 6493  Atomscatm 39726  PSubSpcpsubsp 39959  WAtomscwpointsN 40449  PAutcpautN 40450  DilcdilN 40565

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-dilN 40569

This theorem is referenced by:  isdilN  40617